Contribution
A causal, quantitative criterion for emergence. Earlier arguments for emergence were qualitative, and the few quantitative ones (Bar-Yam’s multiscale variety, Seth’s Granger emergence) were not based on causal models. The paper fixes a micro mechanism completely, defines macro descriptions by coarse-graining it in space or time, measures effective information at each level under interventions, and shows by worked examples that the macro can score higher. It names the difference in bits “causal emergence”.
Key insight
EI is effectiveness times size. A finer description has more states, so more room for information, but its mechanisms may be noisy or degenerate. A coarser description has fewer states, but grouping can average out noise and merge states that lead to the same place. When the second effect beats the first, the coarse description is the more informative causal model of the same system, even though every fact about it is fixed by the micro.
Assumptions
- Discrete, finite, time-invariant systems of binary elements, each implementing a logical function of its inputs. The transition probability matrix (TPM) is known exactly. There are no data and no estimation.
- Uniform interventions at each level. The TPM is obtained by setting the system “into all possible n initial states with equal probability 1/n”, Pearl’s do-operator applied to every state. At a macro level the uniform distribution is imposed on macro states, so each macro state’s effect is the average over its micro realizers (“macro perturbation”). This is the step that lets the macro win. The authors say so: the “modified distribution of micro perturbations … makes EI sensitive to the causal structure at each level, ultimately allowing the supervening macro EI to exceed the micro EI”.
- Admissible coarse-grainings. The mapping must be exhaustive and disjoint over micro elements, and “no micro-level information is available at the macro level”: a macro state may not depend on which micro element is in which state. So [00, 01, 10] = off, [11] = on is allowed, and [[00, 01], [10, 11]] is not.
- Supervenience and exclusion are both accepted. No causal ingredient is added at the macro level, and causation is counted at one level only.
- The search over coarse-grainings is exhaustive, done “with a custom-made Python program”, and is said to be infeasible for real systems.
Key results
- Definitions. For a current state s₀, cause information is D_KL(cause repertoire ‖ U_C) and effect information is D_KL(effect repertoire ‖ U_E). Both are bounded by log₂ n. EI(S) is the expected effect information over uniformly imposed states, which equals the mutual information between the intervention distribution and the resulting effects, MI(U_C; U_E).
- Effectiveness. Eff(S) = EI(S)/log₂ n, and Eff = ⟨determinism coefficient⟩ − ⟨degeneracy coefficient⟩. Determinism is 1 when each state leads to one future state with certainty and 0 when every future is equally likely. Degeneracy measures “deterministic convergence (not due to noise)” of different states onto the same future.
- Causal emergence. CE = EI(S_M) − EI(S_m), in bits. CE > 0 is emergence; CE < 0 at every macro level is causal reduction. The gain splits exactly as CE = ΔI_Eff + ΔI_Size, with ΔI_Eff = (Eff(S_M) − Eff(S_m))·log₂ n_M and ΔI_Size = Eff(S_m)·(log₂ n_M − log₂ n_m), which is always negative. Emergence requires ΔI_Eff > −ΔI_Size.
- Four examples.
- Spatial, noise. Four noisy AND gates. EI(S_m) = 1.15 bits (Eff 0.29); grouping into two macro elements gives EI(S_M) = 1.55 bits (Eff 0.78). CE = 0.40 bits, 91% of the effectiveness gain from determinism.
- Spatial, degeneracy. Six deterministic AND gates wired for high degeneracy (degeneracy coefficient 0.6), EI(S_m) = 2.43 bits; three macro COPY gates give EI(S_M) = 3 bits, Eff 1. CE = 0.57 bits.
- Temporal. Second-order Markov elements that answer a two-spike burst with a two-spike burst. One micro step: EI 0.16 bits; two micro steps: 1.38 bits; one macro step over two micro steps: 2 bits, Eff 1. CE = 0.62 bits.
- Spatiotemporal. Nine fully connected, heterogeneous, spontaneously active elements in three groups, read as “minicolumns” with states inhibited, receptive and bursting. EI(S_m) = 0.59 bits, EI(S_M) = 3.51 bits, CE = 2.92 bits, from a rise in determinism that outweighs a slight rise in degeneracy.
- Rules of thumb (Applicability). If Eff(S_m) ≥ Eff(S_M), emergence is impossible. If EI(S_m) > log₂ n_M, reduction holds. A drastic rise in Eff under some coarse-graining makes emergence likely. Systems already near maximal effectiveness at the micro level reduce.
- Interpretive claims (Discussion). The EI-maximising level is called “an intrinsic property of the system”. Exclusion is turned around: “to avoid double-counting causes, optimal macro causation must exclude micro causation”. The maximal level is “optimal to characterize, predict, and retrodict the behavior of the system”, and emergence is said to imply causal “self-definition” at that level.
- Conclusions. Brains are likely to work at a macro level because selection must cope with noise and produces degeneracy, and the hierarchy of the sciences “may not just be a matter of convenience”.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | EI decomposes into effectiveness (determinism minus degeneracy) times log₂ of the state-space size | strong | derivation, with details in the SI (not read) |
| C2 | EI can be higher at a coarse-grained macro level than at the micro level it supervenes on | strong as an existence result | four exactly computed toy systems |
| C3 | The gain arises when coarse-graining raises determinism or lowers degeneracy more than the state space shrinks | strong | the exact decomposition CE = ΔI_Eff + ΔI_Size |
| C4 | The EI-maximising level is intrinsic to the system and is where its causal work is done | weak | argued, given the uniform intervention distribution, which is a choice the paper does not test |
| C5 | Macro causation, when it wins, excludes micro causation | weak | an application of Kim’s exclusion principle to C4, not argued separately |
| C6 | Brains and other selected systems are likely to work at a macro level | speculative | an argument from noise and degeneracy in selected systems; no measurement |
Concepts
- Effective information (EI). Mutual information between a uniform distribution of interventions over a system’s states and the resulting next states. A property of the TPM, not of observed data.
- Effectiveness (Eff). EI divided by log₂ n; 1 for a deterministic, non-degenerate (reversible) system, 0 for a fully noisy or fully degenerate one.
- Determinism. How reliably a state leads to one future state.
- Degeneracy. How far different states deterministically lead to the same future state.
- Macro perturbation. Setting a macro state by setting each of its micro realizers with equal probability and averaging the effects.
- Causal emergence (CE). EI(S_M) − EI(S_m) at the optimal macro level; positive means the macro supersedes the micro.
- Supervenience versus supersedence. The macro is fixed by the micro (supervenience) but may carry more EI (supersedence).
Connections
Built on Tononi & Sporns (2003) for EI, Pearl (2000) for interventions, Tononi, Sporns & Edelman (1999) for degeneracy, and Kim (1993, 2000) for the exclusion argument it turns around. It names channel capacity as the parallel: both search over input distributions, but here the system is its own channel and the distributions must come from admissible coarse-grainings. It names Markov-chain lumpability and ε-machines as future work.
In the record, later work in the same line is Varley & Hoel (LIT-027), which re-describes EI through a partial information decomposition, and the ΦID account reviewed in LIT-025, which cites this paper as a “not too dissimilar” approach (NOTE-010). Levin (LIT-439, NOTE-341) and Rizi (NOTE-153) name it as the way to compare levels without using it.
Bearing on the record
On QUESTION-093: an interventional criterion, met without closure or self-individuation.
- EI is defined by interventions, not by prediction from observed statistics. So CE > 0 is a measure of what the question calls interventional autonomy: higher-level variables that support more informative predictions about the effects of interventions. The paper does claim the EI-optimal level is also best “to characterize, predict, and retrodict”, so it bundles predictive autonomy in, but only for predictions from imposed uniform states, not from the system’s own history.
- None of the paper’s systems maintains itself. They are logic gates with a fixed TPM. Positive CE therefore does not need organisational closure.
- The coarse-graining is chosen by an exhaustive search the analyst runs. The paper calls the optimum “intrinsic”, meaning that the EI maximum is fixed by the TPM once the admissible mappings and the uniform intervention distribution are fixed. That is not self-individuation in THEORY-071’s sense, where the system distinguishes itself from its environment. The paper’s “self-definition” is a figure of speech for the EI maximum.
- So the paper is evidence that the first criteria can be had without the last two, which is half of what the question asks. It is not evidence about whether closure or self-individuation can be had without them.
On the reification rule (CLAIM-154) and QUESTION-040: the same shape of trade-off, for a different question.
- CE = ΔI_Eff + ΔI_Size has the form of CLAIM-154’s ½(d’² − Occam): a gain set against a size cost that is always paid. The resemblance stops there. CLAIM-154 asks whether data justify keeping a candidate structure, and its cost is the information the data add about it. Hoel et al. assume the mechanism known exactly. Their cost is the loss of states under coarse-graining, and there is no prior, no posterior and no sample. The two can be combined: CLAIM-154 decides whether a coarse block is supported by data, and CE says whether, granted the mechanism, the coarse description is the more effective causal model. Neither implies the other. This is a complement, not a rival and not a case.
- QUESTION-040 asks how far searching for a candidate inflates its score. The paper searches all admissible coarse-grainings and reports the maximum. With an exact TPM there is nothing to inflate. With an estimated TPM, which every application to a real system would need, the maximum over many coarse-grainings would be biased upward, and the paper does not discuss it. The question applies to this measure unchanged.
On proportional causation (THEORY-133): a complement that supplies a number.
- List & Menzies (LIT-722) hold that a higher-level property excludes its realizer as a cause when the effect would survive a change of realizer. Hoel et al. accept Kim’s exclusion principle and turn it around: when the macro wins, it “must exclude micro causation”. Both reach downward exclusion system by system, not a priori.
- They reach it differently. List & Menzies use counterfactuals over nearest worlds; Hoel et al. average effects over all micro realizers of a macro state and compare information. Degeneracy reduction, one of the two routes to CE, is the information-theoretic face of realization insensitivity: many micro states lead to the same macro effect.
- Neither paper cites the other. The comparison is the record’s.
On the information-decomposition account (LIT-025): a different formal object under the same name. EI is interventional and uses a uniform distribution imposed on the states; ΦID emergence is a property of a supervenient variable’s unique information about the future under the system’s own dynamics. One compares a macro model against a micro model; the other asks what a macro feature predicts that no part does. NOTE-010 already records the review’s claim that the two are compatible. They can disagree on a given system, and the record holds no case that tests them together. No rivalry is declared, since the record holds no THEORY for either.
Note of 2026-10-16: THEORY-222 now cites LIT-1038 as a source.
Limitations
- The intervention distribution carries the result. Uniform interventions are imposed independently at each level, so the macro is scored under a different distribution of micro states than the micro. The paper presents this as what makes the comparison fair. Whether a different choice, such as the system’s stationary distribution, would keep the macro ahead is not tested.
- Toy systems only. Four hand-built examples, each designed to show emergence (Fig. S1, a case of reduction, is in the unread SI). No real system is measured.
- Exhaustive search. Infeasible beyond a few elements, by the authors’ own account. The rules of thumb are offered instead.
- EI rewards determinism. A system can raise EI by being reversible. Whether more EI means more causation, rather than more invertibility, is assumed.
- “Intrinsic” depends on two choices: the admissible mappings and the intervention distribution.
- The claims about brains and the sciences in the Conclusions are conjecture.
Open questions
- Does the macro advantage survive when the intervention distribution is the system’s own stationary distribution, or when the TPM is estimated from data with a correction for the search?
- On a system that also has closure, such as a protocell model, does the EI-optimal coarse-graining coincide with the boundary the organisation maintains? That would bear on whether QUESTION-093’s criteria converge.
- Do EI, the ΦID criteria and CLAIM-154’s rule pick the same level on one shared model?